'n' persons meet in a room and each shakes hands with all the others. lf the total numbers of hand - shakes is 21 0, then 'n' is
n*(n-1)/2=210
n^2-n-420=0
n=(1+-sqrt(1+1680)/2=.5+20.5=21
Now compare this to
21+20+19+......+2+1
this is an arithmetic series, a1=20, r=-1
The sum of an arithmetic series is found by multiplying the number of terms times the average of the first and last terms.
sum=(21+1)/2 *21=11*20=210